- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001728: Permutations ⟶ ℤ

Values

[1,0] => [1] => [1] => 0

[1,0,1,0] => [1,2] => [1,2] => 0

[1,1,0,0] => [2,1] => [2,1] => 0

[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0

[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 0

[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 0

[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 1

[1,1,1,0,0,0] => [3,2,1] => [3,1,2] => 0

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 0

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 0

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => 1

[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,2,3] => 0

[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 0

[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 0

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => 1

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => 1

[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,2,1,3] => 1

[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,1,2,4] => 0

[1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,3,2,1] => 1

[1,1,1,0,1,0,0,0] => [3,4,2,1] => [4,1,3,2] => 0

[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,1,2,3] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 0

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 0

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => 1

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,5,3,4] => 0

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 0

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 0

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => 1

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => 1

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,5,3,2,4] => 1

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,2,3,5] => 0

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,4,3,2] => 1

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,5,2,4,3] => 0

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,2,3,4] => 0

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 0

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 0

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 0

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => 1

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,5,3,4] => 0

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => 1

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => 1

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,2,3,1,4] => 1

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,2,1,3,5] => 1

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [5,2,4,3,1] => 1

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [5,2,1,4,3] => 1

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,2,1,3,4] => 1

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,1,2,4,5] => 0

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,1,2,5,4] => 0

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [4,3,2,1,5] => 1

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [5,3,2,4,1] => 1

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [5,3,2,1,4] => 1

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [4,1,3,2,5] => 0

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,4,3,2,1] => 2

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [5,1,3,4,2] => 0

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [5,1,3,2,4] => 0

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,1,2,3,5] => 0

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,4,2,3,1] => 1

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [5,1,4,3,2] => 0

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [5,1,2,4,3] => 0

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,1,2,3,4] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 0

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 0

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 1

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,6,4,5] => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,6,4,3,5] => 1

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,5,3,4,6] => 0

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,6,5,4,3] => 1

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,6,3,5,4] => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 0

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 0

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 0

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => 1

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,6,4,5] => 0

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 1

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => 1

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,6,3,4,2,5] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,5,3,2,4,6] => 1

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,6,3,5,4,2] => 1

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,6,3,2,5,4] => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,6,3,2,4,5] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,2,3,5,6] => 0

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,4,2,3,6,5] => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,5,4,3,2,6] => 1

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,6,4,3,5,2] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,6,4,3,2,5] => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,5,2,4,3,6] => 0

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,6,5,4,3,2] => 2

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,6,2,4,5,3] => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,6,2,4,3,5] => 0

>>> Load all 262 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 5\ q$

$F_{5} = 22 + 19\ q + q^{2}$

$F_{6} = 57 + 66\ q + 9\ q^{2}$

Description

The number of invisible descents of a permutation.
A visible descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i+1) \leq \min(i, \pi(i))$ . Thus, an invisible descent satisfies

$\pi(i) > \pi(i+1) > i$ .

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.